LMFDB - Embedded newform 1045.4.a.g.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1045,4,Mod(1,1045)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1045, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1045.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1045 = 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1045.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$61.6569959560$
Analytic rank:	$0$
Dimension:	$23$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Character	$\chi$	$=$	1045.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.51379 q^{2} +1.98888 q^{3} -5.70843 q^{4} -5.00000 q^{5} +3.01075 q^{6} -28.4418 q^{7} -20.7517 q^{8} -23.0444 q^{9} +O(q^{10})$	\(q+1.51379 q^{2} +1.98888 q^{3} -5.70843 q^{4} -5.00000 q^{5} +3.01075 q^{6} -28.4418 q^{7} -20.7517 q^{8} -23.0444 q^{9} -7.56896 q^{10} -11.0000 q^{11} -11.3534 q^{12} -91.9365 q^{13} -43.0550 q^{14} -9.94440 q^{15} +14.2537 q^{16} +29.0687 q^{17} -34.8844 q^{18} -19.0000 q^{19} +28.5422 q^{20} -56.5673 q^{21} -16.6517 q^{22} +118.567 q^{23} -41.2727 q^{24} +25.0000 q^{25} -139.173 q^{26} -99.5322 q^{27} +162.358 q^{28} -170.837 q^{29} -15.0537 q^{30} -189.923 q^{31} +187.591 q^{32} -21.8777 q^{33} +44.0039 q^{34} +142.209 q^{35} +131.547 q^{36} +218.344 q^{37} -28.7620 q^{38} -182.851 q^{39} +103.759 q^{40} +105.055 q^{41} -85.6311 q^{42} +108.424 q^{43} +62.7928 q^{44} +115.222 q^{45} +179.486 q^{46} +61.3255 q^{47} +28.3489 q^{48} +465.936 q^{49} +37.8448 q^{50} +57.8140 q^{51} +524.814 q^{52} -398.893 q^{53} -150.671 q^{54} +55.0000 q^{55} +590.216 q^{56} -37.7887 q^{57} -258.612 q^{58} +238.777 q^{59} +56.7669 q^{60} +148.090 q^{61} -287.503 q^{62} +655.423 q^{63} +169.944 q^{64} +459.683 q^{65} -33.1182 q^{66} -796.356 q^{67} -165.936 q^{68} +235.815 q^{69} +215.275 q^{70} +535.775 q^{71} +478.210 q^{72} -777.904 q^{73} +330.527 q^{74} +49.7220 q^{75} +108.460 q^{76} +312.860 q^{77} -276.798 q^{78} -768.112 q^{79} -71.2685 q^{80} +424.240 q^{81} +159.031 q^{82} -612.703 q^{83} +322.911 q^{84} -145.343 q^{85} +164.132 q^{86} -339.774 q^{87} +228.269 q^{88} -862.046 q^{89} +174.422 q^{90} +2614.84 q^{91} -676.831 q^{92} -377.733 q^{93} +92.8341 q^{94} +95.0000 q^{95} +373.096 q^{96} +1446.62 q^{97} +705.331 q^{98} +253.488 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q + 6 q^{2} + 9 q^{3} + 92 q^{4} - 115 q^{5} - 25 q^{6} - 37 q^{7} + 42 q^{8} + 170 q^{9}+O(q^{10}) $ 23 * q + 6 * q^2 + 9 * q^3 + 92 * q^4 - 115 * q^5 - 25 * q^6 - 37 * q^7 + 42 * q^8 + 170 * q^9	\( 23 q + 6 q^{2} + 9 q^{3} + 92 q^{4} - 115 q^{5} - 25 q^{6} - 37 q^{7} + 42 q^{8} + 170 q^{9} - 30 q^{10} - 253 q^{11} + 44 q^{12} - 37 q^{13} + 61 q^{14} - 45 q^{15} + 588 q^{16} - 73 q^{17} + 391 q^{18} - 437 q^{19} - 460 q^{20} - 127 q^{21} - 66 q^{22} - 175 q^{23} + 16 q^{24} + 575 q^{25} + 719 q^{26} + 21 q^{27} + 253 q^{28} + 71 q^{29} + 125 q^{30} + 302 q^{31} + 1107 q^{32} - 99 q^{33} + 1267 q^{34} + 185 q^{35} + 703 q^{36} - 500 q^{37} - 114 q^{38} + 457 q^{39} - 210 q^{40} + 770 q^{41} + 2596 q^{42} - 902 q^{43} - 1012 q^{44} - 850 q^{45} - 1101 q^{46} + 356 q^{47} + 1221 q^{48} + 908 q^{49} + 150 q^{50} - 451 q^{51} - 358 q^{52} + 1327 q^{53} + 2534 q^{54} + 1265 q^{55} + 3135 q^{56} - 171 q^{57} + 1014 q^{58} + 3619 q^{59} - 220 q^{60} - 1432 q^{61} + 1826 q^{62} + 1658 q^{63} + 4006 q^{64} + 185 q^{65} + 275 q^{66} - 605 q^{67} + 5128 q^{68} + 3099 q^{69} - 305 q^{70} + 3230 q^{71} + 2152 q^{72} - 637 q^{73} + 5063 q^{74} + 225 q^{75} - 1748 q^{76} + 407 q^{77} + 7230 q^{78} + 2074 q^{79} - 2940 q^{80} + 2291 q^{81} + 530 q^{82} + 3882 q^{83} + 5096 q^{84} + 365 q^{85} + 2262 q^{86} - 27 q^{87} - 462 q^{88} - 210 q^{89} - 1955 q^{90} + 4133 q^{91} - 6064 q^{92} + 824 q^{93} - 392 q^{94} + 2185 q^{95} + 2462 q^{96} + 2032 q^{97} + 7896 q^{98} - 1870 q^{99}+O(q^{100}) \) 23 * q + 6 * q^2 + 9 * q^3 + 92 * q^4 - 115 * q^5 - 25 * q^6 - 37 * q^7 + 42 * q^8 + 170 * q^9 - 30 * q^10 - 253 * q^11 + 44 * q^12 - 37 * q^13 + 61 * q^14 - 45 * q^15 + 588 * q^16 - 73 * q^17 + 391 * q^18 - 437 * q^19 - 460 * q^20 - 127 * q^21 - 66 * q^22 - 175 * q^23 + 16 * q^24 + 575 * q^25 + 719 * q^26 + 21 * q^27 + 253 * q^28 + 71 * q^29 + 125 * q^30 + 302 * q^31 + 1107 * q^32 - 99 * q^33 + 1267 * q^34 + 185 * q^35 + 703 * q^36 - 500 * q^37 - 114 * q^38 + 457 * q^39 - 210 * q^40 + 770 * q^41 + 2596 * q^42 - 902 * q^43 - 1012 * q^44 - 850 * q^45 - 1101 * q^46 + 356 * q^47 + 1221 * q^48 + 908 * q^49 + 150 * q^50 - 451 * q^51 - 358 * q^52 + 1327 * q^53 + 2534 * q^54 + 1265 * q^55 + 3135 * q^56 - 171 * q^57 + 1014 * q^58 + 3619 * q^59 - 220 * q^60 - 1432 * q^61 + 1826 * q^62 + 1658 * q^63 + 4006 * q^64 + 185 * q^65 + 275 * q^66 - 605 * q^67 + 5128 * q^68 + 3099 * q^69 - 305 * q^70 + 3230 * q^71 + 2152 * q^72 - 637 * q^73 + 5063 * q^74 + 225 * q^75 - 1748 * q^76 + 407 * q^77 + 7230 * q^78 + 2074 * q^79 - 2940 * q^80 + 2291 * q^81 + 530 * q^82 + 3882 * q^83 + 5096 * q^84 + 365 * q^85 + 2262 * q^86 - 27 * q^87 - 462 * q^88 - 210 * q^89 - 1955 * q^90 + 4133 * q^91 - 6064 * q^92 + 824 * q^93 - 392 * q^94 + 2185 * q^95 + 2462 * q^96 + 2032 * q^97 + 7896 * q^98 - 1870 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.51379	0.535206	0.267603	−	0.963529i	$-0.413768\pi$
$2$	1.51379	0.535206	0.267603	+	0.963529i	$0.413768\pi$
$3$	1.98888	0.382760	0.191380	−	0.981516i	$-0.438704\pi$
$3$	1.98888	0.382760	0.191380	+	0.981516i	$0.438704\pi$
$4$	−5.70843	−0.713554
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	3.01075	0.204856
$7$	−28.4418	−1.53571	−0.767857	−	0.640622i	$-0.778677\pi$
$7$	−28.4418	−1.53571	−0.767857	+	0.640622i	$0.778677\pi$
$8$	−20.7517	−0.917105
$9$	−23.0444	−0.853495
$10$	−7.56896	−0.239352
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	−11.3534	−0.273120
$13$	−91.9365	−1.96143	−0.980716	−	0.195440i	$-0.937387\pi$
$13$	−91.9365	−1.96143	−0.980716	+	0.195440i	$0.937387\pi$
$14$	−43.0550	−0.821923
$15$	−9.94440	−0.171175
$16$	14.2537	0.222714
$17$	29.0687	0.414717	0.207358	−	0.978265i	$-0.433513\pi$
$17$	29.0687	0.414717	0.207358	+	0.978265i	$0.433513\pi$
$18$	−34.8844	−0.456796
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	28.5422	0.319111
$21$	−56.5673	−0.587810
$22$	−16.6517	−0.161371
$23$	118.567	1.07491	0.537454	−	0.843293i	$-0.319386\pi$
$23$	118.567	1.07491	0.537454	+	0.843293i	$0.319386\pi$
$24$	−41.2727	−0.351031
$25$	25.0000	0.200000
$26$	−139.173	−1.04977
$27$	−99.5322	−0.709444
$28$	162.358	1.09581
$29$	−170.837	−1.09392	−0.546959	−	0.837159i	$-0.684215\pi$
$29$	−170.837	−1.09392	−0.546959	+	0.837159i	$0.684215\pi$
$30$	−15.0537	−0.0916142
$31$	−189.923	−1.10036	−0.550179	−	0.835047i	$-0.685441\pi$
$31$	−189.923	−1.10036	−0.550179	+	0.835047i	$0.685441\pi$
$32$	187.591	1.03630
$33$	−21.8777	−0.115406
$34$	44.0039	0.221959
$35$	142.209	0.686792
$36$	131.547	0.609015
$37$	218.344	0.970149	0.485074	−	0.874473i	$-0.338793\pi$
$37$	218.344	0.970149	0.485074	+	0.874473i	$0.338793\pi$
$38$	−28.7620	−0.122785
$39$	−182.851	−0.750757
$40$	103.759	0.410142
$41$	105.055	0.400167	0.200083	−	0.979779i	$-0.435879\pi$
$41$	105.055	0.400167	0.200083	+	0.979779i	$0.435879\pi$
$42$	−85.6311	−0.314599
$43$	108.424	0.384524	0.192262	−	0.981344i	$-0.438418\pi$
$43$	108.424	0.384524	0.192262	+	0.981344i	$0.438418\pi$
$44$	62.7928	0.215145
$45$	115.222	0.381694
$46$	179.486	0.575298
$47$	61.3255	0.190324	0.0951622	−	0.995462i	$-0.469663\pi$
$47$	61.3255	0.190324	0.0951622	+	0.995462i	$0.469663\pi$
$48$	28.3489	0.0852460
$49$	465.936	1.35841
$50$	37.8448	0.107041
$51$	57.8140	0.158737
$52$	524.814	1.39959
$53$	−398.893	−1.03381	−0.516907	−	0.856041i	$-0.672917\pi$
$53$	−398.893	−1.03381	−0.516907	+	0.856041i	$0.672917\pi$
$54$	−150.671	−0.379699
$55$	55.0000	0.134840
$56$	590.216	1.40841
$57$	−37.7887	−0.0878112
$58$	−258.612	−0.585472
$59$	238.777	0.526884	0.263442	−	0.964675i	$-0.415142\pi$
$59$	238.777	0.526884	0.263442	+	0.964675i	$0.415142\pi$
$60$	56.7669	0.122143
$61$	148.090	0.310836	0.155418	−	0.987849i	$-0.450328\pi$
$61$	148.090	0.310836	0.155418	+	0.987849i	$0.450328\pi$
$62$	−287.503	−0.588919
$63$	655.423	1.31072
$64$	169.944	0.331922
$65$	459.683	0.877179
$66$	−33.1182	−0.0617663
$67$	−796.356	−1.45210	−0.726048	−	0.687644i	$-0.758645\pi$
$67$	−796.356	−1.45210	−0.726048	+	0.687644i	$0.758645\pi$
$68$	−165.936	−0.295923
$69$	235.815	0.411432
$70$	215.275	0.367575
$71$	535.775	0.895560	0.447780	−	0.894144i	$-0.352215\pi$
$71$	535.775	0.895560	0.447780	+	0.894144i	$0.352215\pi$
$72$	478.210	0.782744
$73$	−777.904	−1.24722	−0.623608	−	0.781737i	$-0.714334\pi$
$73$	−777.904	−1.24722	−0.623608	+	0.781737i	$0.714334\pi$
$74$	330.527	0.519230
$75$	49.7220	0.0765520
$76$	108.460	0.163701
$77$	312.860	0.463035
$78$	−276.798	−0.401810
$79$	−768.112	−1.09392	−0.546958	−	0.837160i	$-0.684214\pi$
$79$	−768.112	−1.09392	−0.546958	+	0.837160i	$0.684214\pi$
$80$	−71.2685	−0.0996007
$81$	424.240	0.581948
$82$	159.031	0.214172
$83$	−612.703	−0.810276	−0.405138	−	0.914255i	$-0.632777\pi$
$83$	−612.703	−0.810276	−0.405138	+	0.914255i	$0.632777\pi$
$84$	322.911	0.419434
$85$	−145.343	−0.185467
$86$	164.132	0.205800
$87$	−339.774	−0.418708
$88$	228.269	0.276518
$89$	−862.046	−1.02670	−0.513352	−	0.858178i	$-0.671597\pi$
$89$	−862.046	−1.02670	−0.513352	+	0.858178i	$0.671597\pi$
$90$	174.422	0.204285
$91$	2614.84	3.01220
$92$	−676.831	−0.767006
$93$	−377.733	−0.421173
$94$	92.8341	0.101863
$95$	95.0000	0.102598
$96$	373.096	0.396655
$97$	1446.62	1.51424	0.757122	−	0.653274i	$-0.226605\pi$
$97$	1446.62	1.51424	0.757122	+	0.653274i	$0.226605\pi$
$98$	705.331	0.727032
$99$	253.488	0.257338
$100$	−142.711	−0.142711
$101$	−1222.48	−1.20437	−0.602184	−	0.798357i	$-0.705703\pi$
$101$	−1222.48	−1.20437	−0.602184	+	0.798357i	$0.705703\pi$
$102$	87.5184	0.0849570
$103$	380.483	0.363981	0.181991	−	0.983300i	$-0.441746\pi$
$103$	380.483	0.363981	0.181991	+	0.983300i	$0.441746\pi$
$104$	1907.84	1.79884
$105$	282.837	0.262876
$106$	−603.841	−0.553304
$107$	−1301.51	−1.17590	−0.587951	−	0.808897i	$-0.700065\pi$
$107$	−1301.51	−1.17590	−0.587951	+	0.808897i	$0.700065\pi$
$108$	568.173	0.506227
$109$	982.512	0.863372	0.431686	−	0.902024i	$-0.357919\pi$
$109$	982.512	0.863372	0.431686	+	0.902024i	$0.357919\pi$
$110$	83.2586	0.0721672
$111$	434.259	0.371334
$112$	−405.401	−0.342025
$113$	−1793.94	−1.49345	−0.746726	−	0.665132i	$-0.768375\pi$
$113$	−1793.94	−1.49345	−0.746726	+	0.665132i	$0.768375\pi$
$114$	−57.2042	−0.0469971
$115$	−592.834	−0.480714
$116$	975.212	0.780570
$117$	2118.62	1.67407
$118$	361.459	0.281992
$119$	−826.765	−0.636886
$120$	206.363	0.156986
$121$	121.000	0.0909091
$122$	224.178	0.166361
$123$	208.942	0.153168
$124$	1084.16	0.785166
$125$	−125.000	−0.0894427
$126$	992.174	0.701507
$127$	1517.30	1.06015	0.530075	−	0.847951i	$-0.322164\pi$
$127$	1517.30	1.06015	0.530075	+	0.847951i	$0.322164\pi$
$128$	−1243.47	−0.858656
$129$	215.643	0.147180
$130$	695.864	0.469472
$131$	−409.493	−0.273111	−0.136556	−	0.990632i	$-0.543603\pi$
$131$	−409.493	−0.273111	−0.136556	+	0.990632i	$0.543603\pi$
$132$	124.887	0.0823488
$133$	540.394	0.352317
$134$	−1205.52	−0.777170
$135$	497.661	0.317273
$136$	−603.224	−0.380339
$137$	923.020	0.575612	0.287806	−	0.957689i	$-0.407074\pi$
$137$	923.020	0.575612	0.287806	+	0.957689i	$0.407074\pi$
$138$	356.975	0.220201
$139$	−2704.29	−1.65018	−0.825089	−	0.565003i	$-0.808875\pi$
$139$	−2704.29	−1.65018	−0.825089	+	0.565003i	$0.808875\pi$
$140$	−811.791	−0.490063
$141$	121.969	0.0728485
$142$	811.051	0.479309
$143$	1011.30	0.591394
$144$	−328.467	−0.190085
$145$	854.185	0.489215
$146$	−1177.58	−0.667518
$147$	926.691	0.519947
$148$	−1246.40	−0.692254
$149$	584.495	0.321367	0.160684	−	0.987006i	$-0.448630\pi$
$149$	584.495	0.321367	0.160684	+	0.987006i	$0.448630\pi$
$150$	75.2687	0.0409711
$151$	−2350.83	−1.26694	−0.633469	−	0.773768i	$-0.718369\pi$
$151$	−2350.83	−1.26694	−0.633469	+	0.773768i	$0.718369\pi$
$152$	394.283	0.210398
$153$	−669.869	−0.353959
$154$	473.605	0.247819
$155$	949.613	0.492095
$156$	1043.79	0.535706
$157$	454.749	0.231165	0.115583	−	0.993298i	$-0.463126\pi$
$157$	454.749	0.231165	0.115583	+	0.993298i	$0.463126\pi$
$158$	−1162.76	−0.585471
$159$	−793.350	−0.395703
$160$	−937.954	−0.463449
$161$	−3372.26	−1.65075
$162$	642.211	0.311462
$163$	122.692	0.0589569	0.0294785	−	0.999565i	$-0.490615\pi$
$163$	122.692	0.0589569	0.0294785	+	0.999565i	$0.490615\pi$
$164$	−599.700	−0.285541
$165$	109.388	0.0516113
$166$	−927.505	−0.433665
$167$	−1098.65	−0.509077	−0.254538	−	0.967063i	$-0.581924\pi$
$167$	−1098.65	−0.509077	−0.254538	+	0.967063i	$0.581924\pi$
$168$	1173.87	0.539083
$169$	6255.33	2.84721
$170$	−220.019	−0.0992631
$171$	437.843	0.195805
$172$	−618.932	−0.274379
$173$	3627.22	1.59406	0.797029	−	0.603941i	$-0.206404\pi$
$173$	3627.22	1.59406	0.797029	+	0.603941i	$0.206404\pi$
$174$	−514.347	−0.224095
$175$	−711.045	−0.307143
$176$	−156.791	−0.0671508
$177$	474.899	0.201670
$178$	−1304.96	−0.549499
$179$	2990.19	1.24859	0.624294	−	0.781190i	$-0.285387\pi$
$179$	2990.19	1.24859	0.624294	+	0.781190i	$0.285387\pi$
$180$	−657.736	−0.272360
$181$	851.776	0.349790	0.174895	−	0.984587i	$-0.444041\pi$
$181$	851.776	0.349790	0.174895	+	0.984587i	$0.444041\pi$
$182$	3958.33	1.61215
$183$	294.533	0.118976
$184$	−2460.47	−0.985804
$185$	−1091.72	−0.433864
$186$	−571.810	−0.225415
$187$	−319.755	−0.125042
$188$	−350.073	−0.135807
$189$	2830.88	1.08950
$190$	143.810	0.0549110
$191$	−3305.35	−1.25218	−0.626091	−	0.779750i	$-0.715346\pi$
$191$	−3305.35	−1.25218	−0.626091	+	0.779750i	$0.715346\pi$
$192$	337.998	0.127046
$193$	−1236.69	−0.461237	−0.230618	−	0.973044i	$-0.574075\pi$
$193$	−1236.69	−0.461237	−0.230618	+	0.973044i	$0.574075\pi$
$194$	2189.88	0.810433
$195$	914.253	0.335749
$196$	−2659.77	−0.969303
$197$	682.431	0.246808	0.123404	−	0.992357i	$-0.460619\pi$
$197$	682.431	0.246808	0.123404	+	0.992357i	$0.460619\pi$
$198$	383.728	0.137729
$199$	1256.67	0.447654	0.223827	−	0.974629i	$-0.428145\pi$
$199$	1256.67	0.447654	0.223827	+	0.974629i	$0.428145\pi$
$200$	−518.793	−0.183421
$201$	−1583.86	−0.555804
$202$	−1850.58	−0.644586
$203$	4858.91	1.67994
$204$	−330.028	−0.113267
$205$	−525.275	−0.178960
$206$	575.972	0.194805
$207$	−2732.30	−0.917429
$208$	−1310.44	−0.436838
$209$	209.000	0.0691714
$210$	428.156	0.140693
$211$	4628.04	1.50999	0.754993	−	0.655733i	$-0.227640\pi$
$211$	4628.04	1.50999	0.754993	+	0.655733i	$0.227640\pi$
$212$	2277.05	0.737683
$213$	1065.59	0.342785
$214$	−1970.21	−0.629350
$215$	−542.121	−0.171964
$216$	2065.46	0.650634
$217$	5401.74	1.68984
$218$	1487.32	0.462082
$219$	−1547.16	−0.477384
$220$	−313.964	−0.0962156
$221$	−2672.47	−0.813438
$222$	657.378	0.198740
$223$	−4885.46	−1.46706	−0.733530	−	0.679657i	$-0.762129\pi$
$223$	−4885.46	−1.46706	−0.733530	+	0.679657i	$0.762129\pi$
$224$	−5335.42	−1.59146
$225$	−576.109	−0.170699
$226$	−2715.66	−0.799305
$227$	2024.12	0.591830	0.295915	−	0.955214i	$-0.404375\pi$
$227$	2024.12	0.591830	0.295915	+	0.955214i	$0.404375\pi$
$228$	215.714	0.0626580
$229$	406.555	0.117318	0.0586592	−	0.998278i	$-0.481317\pi$
$229$	406.555	0.117318	0.0586592	+	0.998278i	$0.481317\pi$
$230$	−897.428	−0.257281
$231$	622.240	0.177231
$232$	3545.16	1.00324
$233$	−5434.56	−1.52802	−0.764012	−	0.645202i	$-0.776773\pi$
$233$	−5434.56	−1.52802	−0.764012	+	0.645202i	$0.776773\pi$
$234$	3207.15	0.895974
$235$	−306.628	−0.0851156
$236$	−1363.04	−0.375960
$237$	−1527.68	−0.418707
$238$	−1251.55	−0.340865
$239$	−1663.80	−0.450302	−0.225151	−	0.974324i	$-0.572288\pi$
$239$	−1663.80	−0.450302	−0.225151	+	0.974324i	$0.572288\pi$
$240$	−141.744	−0.0381232
$241$	6333.77	1.69292	0.846460	−	0.532452i	$-0.178729\pi$
$241$	6333.77	1.69292	0.846460	+	0.532452i	$0.178729\pi$
$242$	183.169	0.0486551
$243$	3531.13	0.932190
$244$	−845.363	−0.221798
$245$	−2329.68	−0.607502
$246$	316.294	0.0819764
$247$	1746.79	0.449983
$248$	3941.22	1.00914
$249$	−1218.59	−0.310141
$250$	−189.224	−0.0478703
$251$	6294.40	1.58286	0.791432	−	0.611257i	$-0.209336\pi$
$251$	6294.40	1.58286	0.791432	+	0.611257i	$0.209336\pi$
$252$	−3741.44	−0.935272
$253$	−1304.24	−0.324097
$254$	2296.88	0.567399
$255$	−289.070	−0.0709893
$256$	−3241.90	−0.791480
$257$	3985.46	0.967340	0.483670	−	0.875250i	$-0.339304\pi$
$257$	3985.46	0.967340	0.483670	+	0.875250i	$0.339304\pi$
$258$	326.438	0.0787719
$259$	−6210.09	−1.48987
$260$	−2624.07	−0.625915
$261$	3936.83	0.933653
$262$	−619.887	−0.146171
$263$	−4084.47	−0.957640	−0.478820	−	0.877913i	$-0.658935\pi$
$263$	−4084.47	−0.957640	−0.478820	+	0.877913i	$0.658935\pi$
$264$	453.999	0.105840
$265$	1994.46	0.462336
$266$	818.044	0.188562
$267$	−1714.51	−0.392981
$268$	4545.95	1.03615
$269$	−3515.10	−0.796726	−0.398363	−	0.917228i	$-0.630422\pi$
$269$	−3515.10	−0.796726	−0.398363	+	0.917228i	$0.630422\pi$
$270$	753.355	0.169806
$271$	6091.30	1.36539	0.682694	−	0.730704i	$-0.260808\pi$
$271$	6091.30	1.36539	0.682694	+	0.730704i	$0.260808\pi$
$272$	414.336	0.0923632
$273$	5200.60	1.15295
$274$	1397.26	0.308071
$275$	−275.000	−0.0603023
$276$	−1346.14	−0.293579
$277$	−7122.35	−1.54491	−0.772456	−	0.635069i	$-0.780972\pi$
$277$	−7122.35	−1.54491	−0.772456	+	0.635069i	$0.780972\pi$
$278$	−4093.73	−0.883186
$279$	4376.65	0.939150
$280$	−2951.08	−0.629860
$281$	−3157.23	−0.670265	−0.335133	−	0.942171i	$-0.608781\pi$
$281$	−3157.23	−0.670265	−0.335133	+	0.942171i	$0.608781\pi$
$282$	184.636	0.0389890
$283$	−1370.21	−0.287810	−0.143905	−	0.989591i	$-0.545966\pi$
$283$	−1370.21	−0.287810	−0.143905	+	0.989591i	$0.545966\pi$
$284$	−3058.43	−0.639031
$285$	188.944	0.0392703
$286$	1530.90	0.316518
$287$	−2987.95	−0.614541
$288$	−4322.91	−0.884479
$289$	−4068.01	−0.828010
$290$	1293.06	0.261831
$291$	2877.15	0.579592
$292$	4440.61	0.889956
$293$	4409.32	0.879165	0.439583	−	0.898202i	$-0.355126\pi$
$293$	4409.32	0.879165	0.439583	+	0.898202i	$0.355126\pi$
$294$	1402.82	0.278279
$295$	−1193.89	−0.235630
$296$	−4531.01	−0.889728
$297$	1094.85	0.213905
$298$	884.804	0.171998
$299$	−10900.6	−2.10836
$300$	−283.835	−0.0546240
$301$	−3083.78	−0.590518
$302$	−3558.66	−0.678073
$303$	−2431.36	−0.460984
$304$	−270.820	−0.0510941
$305$	−740.451	−0.139010
$306$	−1014.04	−0.189441
$307$	1801.38	0.334886	0.167443	−	0.985882i	$-0.446449\pi$
$307$	1801.38	0.334886	0.167443	+	0.985882i	$0.446449\pi$
$308$	−1785.94	−0.330401
$309$	756.734	0.139317
$310$	1437.52	0.263373
$311$	−3614.40	−0.659016	−0.329508	−	0.944153i	$-0.606883\pi$
$311$	−3614.40	−0.659016	−0.329508	+	0.944153i	$0.606883\pi$
$312$	3794.47	0.688523
$313$	2081.95	0.375970	0.187985	−	0.982172i	$-0.439804\pi$
$313$	2081.95	0.375970	0.187985	+	0.982172i	$0.439804\pi$
$314$	688.396	0.123721
$315$	−3277.12	−0.586173
$316$	4384.72	0.780569
$317$	−8983.15	−1.59162	−0.795811	−	0.605545i	$-0.792955\pi$
$317$	−8983.15	−1.59162	−0.795811	+	0.605545i	$0.792955\pi$
$318$	−1200.97	−0.211783
$319$	1879.21	0.329829
$320$	−849.720	−0.148440
$321$	−2588.54	−0.450088
$322$	−5104.89	−0.883492
$323$	−552.304	−0.0951425
$324$	−2421.75	−0.415252
$325$	−2298.41	−0.392286
$326$	185.730	0.0315541
$327$	1954.10	0.330464
$328$	−2180.07	−0.366995
$329$	−1744.21	−0.292284
$330$	165.591	0.0276227
$331$	1312.46	0.217944	0.108972	−	0.994045i	$-0.465244\pi$
$331$	1312.46	0.217944	0.108972	+	0.994045i	$0.465244\pi$
$332$	3497.58	0.578176
$333$	−5031.59	−0.828017
$334$	−1663.12	−0.272461
$335$	3981.78	0.649397
$336$	−806.293	−0.130913
$337$	6093.38	0.984948	0.492474	−	0.870327i	$-0.336093\pi$
$337$	6093.38	0.984948	0.492474	+	0.870327i	$0.336093\pi$
$338$	9469.26	1.52385
$339$	−3567.94	−0.571634
$340$	829.682	0.132341
$341$	2089.15	0.331771
$342$	662.803	0.104796
$343$	−3496.53	−0.550422
$344$	−2249.99	−0.352649
$345$	−1179.08	−0.183998
$346$	5490.85	0.853150
$347$	11614.8	1.79688	0.898440	−	0.439096i	$-0.144701\pi$
$347$	11614.8	1.79688	0.898440	+	0.439096i	$0.144701\pi$
$348$	1939.58	0.298771
$349$	1919.96	0.294478	0.147239	−	0.989101i	$-0.452961\pi$
$349$	1919.96	0.294478	0.147239	+	0.989101i	$0.452961\pi$
$350$	−1076.37	−0.164385
$351$	9150.64	1.39152
$352$	−2063.50	−0.312457
$353$	3307.26	0.498661	0.249331	−	0.968418i	$-0.419789\pi$
$353$	3307.26	0.498661	0.249331	+	0.968418i	$0.419789\pi$
$354$	718.898	0.107935
$355$	−2678.87	−0.400507
$356$	4920.93	0.732609
$357$	−1644.34	−0.243774
$358$	4526.52	0.668252
$359$	−3821.65	−0.561836	−0.280918	−	0.959732i	$-0.590639\pi$
$359$	−3821.65	−0.561836	−0.280918	+	0.959732i	$0.590639\pi$
$360$	−2391.05	−0.350054
$361$	361.000	0.0526316
$362$	1289.41	0.187210
$363$	240.654	0.0347964
$364$	−14926.6	−2.14937
$365$	3889.52	0.557772
$366$	445.862	0.0636765
$367$	−96.4881	−0.0137238	−0.00686191	−	0.999976i	$-0.502184\pi$
$367$	−96.4881	−0.0137238	−0.00686191	+	0.999976i	$0.502184\pi$
$368$	1690.02	0.239397
$369$	−2420.93	−0.341540
$370$	−1652.64	−0.232207
$371$	11345.2	1.58764
$372$	2156.27	0.300530
$373$	−2656.14	−0.368712	−0.184356	−	0.982859i	$-0.559020\pi$
$373$	−2656.14	−0.368712	−0.184356	+	0.982859i	$0.559020\pi$
$374$	−484.043	−0.0669232
$375$	−248.610	−0.0342351
$376$	−1272.61	−0.174547
$377$	15706.2	2.14565
$378$	4285.36	0.583108
$379$	6788.20	0.920017	0.460008	−	0.887915i	$-0.347846\pi$
$379$	6788.20	0.920017	0.460008	+	0.887915i	$0.347846\pi$
$380$	−542.301	−0.0732091
$381$	3017.73	0.405783
$382$	−5003.62	−0.670176
$383$	−1228.98	−0.163964	−0.0819819	−	0.996634i	$-0.526125\pi$
$383$	−1228.98	−0.163964	−0.0819819	+	0.996634i	$0.526125\pi$
$384$	−2473.11	−0.328659
$385$	−1564.30	−0.207076
$386$	−1872.09	−0.246857
$387$	−2498.57	−0.328189
$388$	−8257.91	−1.08050
$389$	6810.03	0.887615	0.443807	−	0.896122i	$-0.353627\pi$
$389$	6810.03	0.887615	0.443807	+	0.896122i	$0.353627\pi$
$390$	1383.99	0.179695
$391$	3446.58	0.445783
$392$	−9668.98	−1.24581
$393$	−814.432	−0.104536
$394$	1033.06	0.132093
$395$	3840.56	0.489214
$396$	−1447.02	−0.183625
$397$	7055.68	0.891976	0.445988	−	0.895039i	$-0.352852\pi$
$397$	7055.68	0.891976	0.445988	+	0.895039i	$0.352852\pi$
$398$	1902.34	0.239587
$399$	1074.78	0.134853
$400$	356.342	0.0445428
$401$	4120.61	0.513150	0.256575	−	0.966524i	$-0.417406\pi$
$401$	4120.61	0.513150	0.256575	+	0.966524i	$0.417406\pi$
$402$	−2397.63	−0.297470
$403$	17460.8	2.15828
$404$	6978.44	0.859383
$405$	−2121.20	−0.260255
$406$	7355.38	0.899117
$407$	−2401.78	−0.292511
$408$	−1199.74	−0.145578
$409$	−9137.83	−1.10473	−0.552367	−	0.833601i	$-0.686275\pi$
$409$	−9137.83	−1.10473	−0.552367	+	0.833601i	$0.686275\pi$
$410$	−795.157	−0.0957805
$411$	1835.77	0.220321
$412$	−2171.96	−0.259720
$413$	−6791.26	−0.809143
$414$	−4136.13	−0.491014
$415$	3063.52	0.362367
$416$	−17246.5	−2.03264
$417$	−5378.50	−0.631622
$418$	316.383	0.0370210
$419$	−5239.07	−0.610848	−0.305424	−	0.952216i	$-0.598798\pi$
$419$	−5239.07	−0.610848	−0.305424	+	0.952216i	$0.598798\pi$
$420$	−1614.55	−0.187577
$421$	−3199.17	−0.370352	−0.185176	−	0.982705i	$-0.559285\pi$
$421$	−3199.17	−0.370352	−0.185176	+	0.982705i	$0.559285\pi$
$422$	7005.88	0.808154
$423$	−1413.21	−0.162441
$424$	8277.71	0.948116
$425$	726.716	0.0829433
$426$	1613.08	0.183460
$427$	−4211.95	−0.477355
$428$	7429.57	0.839070
$429$	2011.36	0.226362
$430$	−820.658	−0.0920364
$431$	6295.21	0.703549	0.351774	−	0.936085i	$-0.385578\pi$
$431$	6295.21	0.703549	0.351774	+	0.936085i	$0.385578\pi$
$432$	−1418.70	−0.158003
$433$	15343.0	1.70286	0.851431	−	0.524466i	$-0.175735\pi$
$433$	15343.0	1.70286	0.851431	+	0.524466i	$0.175735\pi$
$434$	8177.12	0.904410
$435$	1698.87	0.187252
$436$	−5608.60	−0.616063
$437$	−2252.77	−0.246601
$438$	−2342.07	−0.255499
$439$	176.799	0.0192213	0.00961067	−	0.999954i	$-0.496941\pi$
$439$	176.799	0.0192213	0.00961067	+	0.999954i	$0.496941\pi$
$440$	−1141.34	−0.123662
$441$	−10737.2	−1.15940
$442$	−4045.57	−0.435357
$443$	−1186.03	−0.127201	−0.0636004	−	0.997975i	$-0.520258\pi$
$443$	−1186.03	−0.127201	−0.0636004	+	0.997975i	$0.520258\pi$
$444$	−2478.94	−0.264967
$445$	4310.23	0.459156
$446$	−7395.57	−0.785180
$447$	1162.49	0.123006
$448$	−4833.51	−0.509737
$449$	−13414.4	−1.40995	−0.704974	−	0.709233i	$-0.749042\pi$
$449$	−13414.4	−1.40995	−0.704974	+	0.709233i	$0.749042\pi$
$450$	−872.109	−0.0913592
$451$	−1155.60	−0.120655
$452$	10240.6	1.06566
$453$	−4675.51	−0.484933
$454$	3064.09	0.316751
$455$	−13074.2	−1.34709
$456$	784.180	0.0805321
$457$	−16120.0	−1.65003	−0.825014	−	0.565112i	$-0.808833\pi$
$457$	−16120.0	−1.65003	−0.825014	+	0.565112i	$0.808833\pi$
$458$	615.440	0.0627895
$459$	−2893.27	−0.294218
$460$	3384.16	0.343015
$461$	−3024.77	−0.305591	−0.152795	−	0.988258i	$-0.548828\pi$
$461$	−3024.77	−0.305591	−0.152795	+	0.988258i	$0.548828\pi$
$462$	941.943	0.0948553
$463$	−9245.42	−0.928015	−0.464008	−	0.885831i	$-0.653589\pi$
$463$	−9245.42	−0.928015	−0.464008	+	0.885831i	$0.653589\pi$
$464$	−2435.06	−0.243631
$465$	1888.67	0.188354
$466$	−8226.79	−0.817808
$467$	3781.70	0.374724	0.187362	−	0.982291i	$-0.440006\pi$
$467$	3781.70	0.374724	0.187362	+	0.982291i	$0.440006\pi$
$468$	−12094.0	−1.19454
$469$	22649.8	2.23000
$470$	−464.170	−0.0455544
$471$	904.441	0.0884808
$472$	−4955.04	−0.483208
$473$	−1192.67	−0.115938
$474$	−2312.59	−0.224095
$475$	−475.000	−0.0458831
$476$	4719.53	0.454453
$477$	9192.23	0.882355
$478$	−2518.65	−0.241005
$479$	−6607.98	−0.630327	−0.315163	−	0.949037i	$-0.602059\pi$
$479$	−6607.98	−0.630327	−0.315163	+	0.949037i	$0.602059\pi$
$480$	−1865.48	−0.177390
$481$	−20073.8	−1.90288
$482$	9588.01	0.906062
$483$	−6707.01	−0.631841
$484$	−690.721	−0.0648686
$485$	−7233.08	−0.677190
$486$	5345.40	0.498914
$487$	−14279.5	−1.32868	−0.664340	−	0.747430i	$-0.731287\pi$
$487$	−14279.5	−1.32868	−0.664340	+	0.747430i	$0.731287\pi$
$488$	−3073.12	−0.285069
$489$	244.020	0.0225664
$490$	−3526.65	−0.325139
$491$	−2990.12	−0.274832	−0.137416	−	0.990513i	$-0.543880\pi$
$491$	−2990.12	−0.274832	−0.137416	+	0.990513i	$0.543880\pi$
$492$	−1192.73	−0.109294
$493$	−4966.00	−0.453666
$494$	2644.28	0.240834
$495$	−1267.44	−0.115085
$496$	−2707.10	−0.245065
$497$	−15238.4	−1.37532
$498$	−1844.70	−0.165990
$499$	18253.1	1.63752	0.818758	−	0.574138i	$-0.194663\pi$
$499$	18253.1	1.63752	0.818758	+	0.574138i	$0.194663\pi$
$500$	713.554	0.0638222
$501$	−2185.08	−0.194854
$502$	9528.41	0.847159
$503$	−14284.3	−1.26622	−0.633109	−	0.774063i	$-0.718221\pi$
$503$	−14284.3	−1.26622	−0.633109	+	0.774063i	$0.718221\pi$
$504$	−13601.2	−1.20207
$505$	6112.40	0.538610
$506$	−1974.34	−0.173459
$507$	12441.1	1.08980
$508$	−8661.43	−0.756474
$509$	−16096.3	−1.40168	−0.700839	−	0.713319i	$-0.747191\pi$
$509$	−16096.3	−1.40168	−0.700839	+	0.713319i	$0.747191\pi$
$510$	−437.592	−0.0379939
$511$	22125.0	1.91537
$512$	5040.17	0.435051
$513$	1891.11	0.162758
$514$	6033.16	0.517726
$515$	−1902.41	−0.162777
$516$	−1230.98	−0.105021
$517$	−674.581	−0.0573850
$518$	−9400.79	−0.797388
$519$	7214.09	0.610142
$520$	−9539.20	−0.804465
$521$	−20985.7	−1.76468	−0.882340	−	0.470613i	$-0.844033\pi$
$521$	−20985.7	−1.76468	−0.882340	+	0.470613i	$0.844033\pi$
$522$	5959.54	0.499697
$523$	−17707.4	−1.48048	−0.740239	−	0.672344i	$-0.765287\pi$
$523$	−17707.4	−1.48048	−0.740239	+	0.672344i	$0.765287\pi$
$524$	2337.56	0.194880
$525$	−1414.18	−0.117562
$526$	−6183.04	−0.512535
$527$	−5520.80	−0.456337
$528$	−311.838	−0.0257026
$529$	1891.10	0.155428
$530$	3019.20	0.247445
$531$	−5502.47	−0.449693
$532$	−3084.81	−0.251397
$533$	−9658.39	−0.784899
$534$	−2595.40	−0.210326
$535$	6507.54	0.525879
$536$	16525.8	1.33172
$537$	5947.13	0.477909
$538$	−5321.13	−0.426413
$539$	−5125.30	−0.409577
$540$	−2840.86	−0.226391
$541$	18590.9	1.47743	0.738713	−	0.674021i	$-0.235434\pi$
$541$	18590.9	1.47743	0.738713	+	0.674021i	$0.235434\pi$
$542$	9220.96	0.730765
$543$	1694.08	0.133886
$544$	5453.01	0.429772
$545$	−4912.56	−0.386112
$546$	7872.63	0.617065
$547$	−16848.8	−1.31701	−0.658505	−	0.752576i	$-0.728811\pi$
$547$	−16848.8	−1.31701	−0.658505	+	0.752576i	$0.728811\pi$
$548$	−5269.00	−0.410731
$549$	−3412.64	−0.265297
$550$	−416.293	−0.0322742
$551$	3245.90	0.250962
$552$	−4893.57	−0.377326
$553$	21846.5	1.67994
$554$	−10781.8	−0.826846
$555$	−2171.30	−0.166066
$556$	15437.3	1.17749
$557$	−17097.3	−1.30060	−0.650301	−	0.759677i	$-0.725357\pi$
$557$	−17097.3	−1.30060	−0.650301	+	0.759677i	$0.725357\pi$
$558$	6625.33	0.502639
$559$	−9968.14	−0.754217
$560$	2027.00	0.152958
$561$	−635.954	−0.0478610
$562$	−4779.39	−0.358730
$563$	20282.5	1.51831	0.759153	−	0.650912i	$-0.225613\pi$
$563$	20282.5	1.51831	0.759153	+	0.650912i	$0.225613\pi$
$564$	−696.252	−0.0519814
$565$	8969.72	0.667892
$566$	−2074.21	−0.154038
$567$	−12066.2	−0.893705
$568$	−11118.2	−0.821323
$569$	−12467.7	−0.918582	−0.459291	−	0.888286i	$-0.651896\pi$
$569$	−12467.7	−0.918582	−0.459291	+	0.888286i	$0.651896\pi$
$570$	286.021	0.0210177
$571$	10052.1	0.736718	0.368359	−	0.929684i	$-0.379920\pi$
$571$	10052.1	0.736718	0.368359	+	0.929684i	$0.379920\pi$
$572$	−5772.95	−0.421992
$573$	−6573.95	−0.479285
$574$	−4523.14	−0.328906
$575$	2964.17	0.214982
$576$	−3916.25	−0.283294
$577$	3652.91	0.263557	0.131779	−	0.991279i	$-0.457931\pi$
$577$	3652.91	0.263557	0.131779	+	0.991279i	$0.457931\pi$
$578$	−6158.13	−0.443156
$579$	−2459.62	−0.176543
$580$	−4876.06	−0.349082
$581$	17426.4	1.24435
$582$	4355.40	0.310201
$583$	4387.82	0.311707
$584$	16142.8	1.14383
$585$	−10593.1	−0.748668
$586$	6674.80	0.470535
$587$	17724.9	1.24631	0.623155	−	0.782098i	$-0.285850\pi$
$587$	17724.9	1.24631	0.623155	+	0.782098i	$0.285850\pi$
$588$	−5289.95	−0.371010
$589$	3608.53	0.252440
$590$	−1807.30	−0.126110
$591$	1357.27	0.0944683
$592$	3112.21	0.216066
$593$	−1321.33	−0.0915016	−0.0457508	−	0.998953i	$-0.514568\pi$
$593$	−1321.33	−0.0915016	−0.0457508	+	0.998953i	$0.514568\pi$
$594$	1657.38	0.114483
$595$	4133.83	0.284824
$596$	−3336.55	−0.229313
$597$	2499.37	0.171344
$598$	−16501.3	−1.12841
$599$	−17531.6	−1.19586	−0.597931	−	0.801548i	$-0.704010\pi$
$599$	−17531.6	−1.19586	−0.597931	+	0.801548i	$0.704010\pi$
$600$	−1031.82	−0.0702062
$601$	20399.5	1.38455	0.692275	−	0.721634i	$-0.256609\pi$
$601$	20399.5	1.38455	0.692275	+	0.721634i	$0.256609\pi$
$602$	−4668.20	−0.316049
$603$	18351.5	1.23936
$604$	13419.5	0.904028
$605$	−605.000	−0.0406558
$606$	−3680.58	−0.246722
$607$	3854.00	0.257709	0.128854	−	0.991664i	$-0.458870\pi$
$607$	3854.00	0.257709	0.128854	+	0.991664i	$0.458870\pi$
$608$	−3564.23	−0.237744
$609$	9663.79	0.643016
$610$	−1120.89	−0.0743991
$611$	−5638.06	−0.373308
$612$	3823.90	0.252569
$613$	476.034	0.0313651	0.0156826	−	0.999877i	$-0.495008\pi$
$613$	476.034	0.0313651	0.0156826	+	0.999877i	$0.495008\pi$
$614$	2726.91	0.179233
$615$	−1044.71	−0.0684987
$616$	−6492.38	−0.424652
$617$	−1494.37	−0.0975058	−0.0487529	−	0.998811i	$-0.515525\pi$
$617$	−1494.37	−0.0975058	−0.0487529	+	0.998811i	$0.515525\pi$
$618$	1145.54	0.0745636
$619$	−7117.81	−0.462179	−0.231090	−	0.972932i	$-0.574229\pi$
$619$	−7117.81	−0.462179	−0.231090	+	0.972932i	$0.574229\pi$
$620$	−5420.81	−0.351137
$621$	−11801.2	−0.762587
$622$	−5471.45	−0.352710
$623$	24518.1	1.57672
$624$	−2606.30	−0.167204
$625$	625.000	0.0400000
$626$	3151.63	0.201221
$627$	415.676	0.0264761
$628$	−2595.91	−0.164949
$629$	6346.96	0.402337
$630$	−4960.87	−0.313724
$631$	1285.58	0.0811065	0.0405532	−	0.999177i	$-0.487088\pi$
$631$	1285.58	0.0811065	0.0405532	+	0.999177i	$0.487088\pi$
$632$	15939.6	1.00324
$633$	9204.60	0.577962
$634$	−13598.6	−0.851846
$635$	−7586.52	−0.474113
$636$	4528.79	0.282355
$637$	−42836.6	−2.66444
$638$	2844.73	0.176526
$639$	−12346.6	−0.764356
$640$	6217.34	0.384003
$641$	25151.0	1.54977	0.774886	−	0.632100i	$-0.217807\pi$
$641$	25151.0	1.54977	0.774886	+	0.632100i	$0.217807\pi$
$642$	−3918.51	−0.240890
$643$	−29155.9	−1.78818	−0.894089	−	0.447890i	$-0.852176\pi$
$643$	−29155.9	−1.78818	−0.894089	+	0.447890i	$0.852176\pi$
$644$	19250.3	1.17790
$645$	−1078.21	−0.0658211
$646$	−836.074	−0.0509209
$647$	−18584.9	−1.12929	−0.564643	−	0.825335i	$-0.690986\pi$
$647$	−18584.9	−1.12929	−0.564643	+	0.825335i	$0.690986\pi$
$648$	−8803.71	−0.533708
$649$	−2626.55	−0.158861
$650$	−3479.32	−0.209954
$651$	10743.4	0.646801
$652$	−700.379	−0.0420690
$653$	−16177.5	−0.969488	−0.484744	−	0.874656i	$-0.661087\pi$
$653$	−16177.5	−0.969488	−0.484744	+	0.874656i	$0.661087\pi$
$654$	2958.10	0.176867
$655$	2047.47	0.122139
$656$	1497.42	0.0891227
$657$	17926.3	1.06449
$658$	−2640.37	−0.156432
$659$	27156.7	1.60527	0.802636	−	0.596470i	$-0.203430\pi$
$659$	27156.7	1.60527	0.802636	+	0.596470i	$0.203430\pi$
$660$	−624.436	−0.0368275
$661$	11426.2	0.672355	0.336178	−	0.941799i	$-0.390866\pi$
$661$	11426.2	0.672355	0.336178	+	0.941799i	$0.390866\pi$
$662$	1986.80	0.116645
$663$	−5315.22	−0.311352
$664$	12714.6	0.743108
$665$	−2701.97	−0.157561
$666$	−7616.78	−0.443160
$667$	−20255.6	−1.17586
$668$	6271.55	0.363254
$669$	−9716.59	−0.561532
$670$	6027.59	0.347561
$671$	−1628.99	−0.0937206
$672$	−10611.5	−0.609149
$673$	6285.75	0.360026	0.180013	−	0.983664i	$-0.442386\pi$
$673$	6285.75	0.360026	0.180013	+	0.983664i	$0.442386\pi$
$674$	9224.11	0.527150
$675$	−2488.30	−0.141889
$676$	−35708.1	−2.03164
$677$	9929.97	0.563722	0.281861	−	0.959455i	$-0.409048\pi$
$677$	9929.97	0.563722	0.281861	+	0.959455i	$0.409048\pi$
$678$	−5401.12	−0.305942
$679$	−41144.4	−2.32544
$680$	3016.12	0.170093
$681$	4025.73	0.226529
$682$	3162.54	0.177566
$683$	−8488.25	−0.475540	−0.237770	−	0.971321i	$-0.576416\pi$
$683$	−8488.25	−0.475540	−0.237770	+	0.971321i	$0.576416\pi$
$684$	−2499.40	−0.139718
$685$	−4615.10	−0.257422
$686$	−5293.02	−0.294589
$687$	808.589	0.0449048
$688$	1545.44	0.0856388
$689$	36672.8	2.02776
$690$	−1784.88	−0.0984769
$691$	18563.7	1.02199	0.510995	−	0.859584i	$-0.329277\pi$
$691$	18563.7	1.02199	0.510995	+	0.859584i	$0.329277\pi$
$692$	−20705.7	−1.13745
$693$	−7209.66	−0.395198
$694$	17582.5	0.961702
$695$	13521.4	0.737982
$696$	7050.90	0.383999
$697$	3053.81	0.165956
$698$	2906.41	0.157606
$699$	−10808.7	−0.584867
$700$	4058.95	0.219163
$701$	32742.9	1.76417	0.882085	−	0.471090i	$-0.156139\pi$
$701$	32742.9	1.76417	0.882085	+	0.471090i	$0.156139\pi$
$702$	13852.2	0.744753
$703$	−4148.53	−0.222567
$704$	−1869.38	−0.100078
$705$	−609.845	−0.0325789
$706$	5006.50	0.266887
$707$	34769.5	1.84957
$708$	−2710.93	−0.143903
$709$	16988.1	0.899863	0.449931	−	0.893063i	$-0.351448\pi$
$709$	16988.1	0.899863	0.449931	+	0.893063i	$0.351448\pi$
$710$	−4055.26	−0.214354
$711$	17700.7	0.933652
$712$	17888.9	0.941596
$713$	−22518.5	−1.18279
$714$	−2489.18	−0.130470
$715$	−5056.51	−0.264479
$716$	−17069.3	−0.890935
$717$	−3309.10	−0.172358
$718$	−5785.18	−0.300698
$719$	10292.3	0.533850	0.266925	−	0.963717i	$-0.413992\pi$
$719$	10292.3	0.533850	0.266925	+	0.963717i	$0.413992\pi$
$720$	1642.34	0.0850087
$721$	−10821.6	−0.558971
$722$	546.479	0.0281688
$723$	12597.1	0.647982
$724$	−4862.31	−0.249594
$725$	−4270.92	−0.218784
$726$	364.301	0.0186232
$727$	8504.00	0.433832	0.216916	−	0.976190i	$-0.430400\pi$
$727$	8504.00	0.433832	0.216916	+	0.976190i	$0.430400\pi$
$728$	−54262.4	−2.76250
$729$	−4431.49	−0.225143
$730$	5887.92	0.298523
$731$	3151.74	0.159468
$732$	−1681.32	−0.0848956
$733$	−26213.4	−1.32089	−0.660445	−	0.750874i	$-0.729632\pi$
$733$	−26213.4	−1.32089	−0.660445	+	0.750874i	$0.729632\pi$
$734$	−146.063	−0.00734507
$735$	−4633.45	−0.232527
$736$	22242.1	1.11393
$737$	8759.92	0.437823
$738$	−3664.78	−0.182794
$739$	7074.72	0.352162	0.176081	−	0.984376i	$-0.443658\pi$
$739$	7074.72	0.352162	0.176081	+	0.984376i	$0.443658\pi$
$740$	6232.01	0.309585
$741$	3474.16	0.172236
$742$	17174.3	0.849716
$743$	12436.9	0.614088	0.307044	−	0.951695i	$-0.400660\pi$
$743$	12436.9	0.614088	0.307044	+	0.951695i	$0.400660\pi$
$744$	7838.61	0.386260
$745$	−2922.47	−0.143720
$746$	−4020.84	−0.197337
$747$	14119.4	0.691567
$748$	1825.30	0.0892241
$749$	37017.2	1.80585
$750$	−376.344	−0.0183228
$751$	−11510.6	−0.559293	−0.279646	−	0.960103i	$-0.590217\pi$
$751$	−11510.6	−0.559293	−0.279646	+	0.960103i	$0.590217\pi$
$752$	874.115	0.0423879
$753$	12518.8	0.605857
$754$	23775.9	1.14836
$755$	11754.1	0.566592
$756$	−16159.9	−0.777419
$757$	20737.5	0.995666	0.497833	−	0.867273i	$-0.334129\pi$
$757$	20737.5	0.995666	0.497833	+	0.867273i	$0.334129\pi$
$758$	10275.9	0.492399
$759$	−2593.97	−0.124051
$760$	−1971.41	−0.0940930
$761$	−22807.9	−1.08645	−0.543223	−	0.839588i	$-0.682796\pi$
$761$	−22807.9	−1.08645	−0.543223	+	0.839588i	$0.682796\pi$
$762$	4568.22	0.217177
$763$	−27944.4	−1.32589
$764$	18868.4	0.893500
$765$	3349.34	0.158295
$766$	−1860.42	−0.0877544
$767$	−21952.4	−1.03345
$768$	−6447.75	−0.302947
$769$	27386.4	1.28424	0.642119	−	0.766605i	$-0.278055\pi$
$769$	27386.4	1.28424	0.642119	+	0.766605i	$0.278055\pi$
$770$	−2368.02	−0.110828
$771$	7926.60	0.370259
$772$	7059.55	0.329117
$773$	−29591.5	−1.37688	−0.688442	−	0.725292i	$-0.741705\pi$
$773$	−29591.5	−1.37688	−0.688442	+	0.725292i	$0.741705\pi$
$774$	−3782.31	−0.175649
$775$	−4748.07	−0.220072
$776$	−30019.8	−1.38872
$777$	−12351.1	−0.570263
$778$	10309.0	0.475057
$779$	−1996.04	−0.0918045
$780$	−5218.95	−0.239575
$781$	−5893.52	−0.270022
$782$	5217.40	0.238586
$783$	17003.8	0.776073
$784$	6641.31	0.302538
$785$	−2273.75	−0.103380
$786$	−1232.88	−0.0559484
$787$	−21768.8	−0.985988	−0.492994	−	0.870033i	$-0.664098\pi$
$787$	−21768.8	−0.985988	−0.492994	+	0.870033i	$0.664098\pi$
$788$	−3895.61	−0.176111
$789$	−8123.52	−0.366546
$790$	5813.81	0.261831
$791$	51023.0	2.29351
$792$	−5260.31	−0.236006
$793$	−13614.9	−0.609684
$794$	10680.8	0.477391
$795$	3966.75	0.176964
$796$	−7173.63	−0.319425
$797$	−11127.7	−0.494561	−0.247280	−	0.968944i	$-0.579537\pi$
$797$	−11127.7	−0.494561	−0.247280	+	0.968944i	$0.579537\pi$
$798$	1626.99	0.0721740
$799$	1782.65	0.0789307
$800$	4689.77	0.207261
$801$	19865.3	0.876287
$802$	6237.74	0.274641
$803$	8556.94	0.376050
$804$	9041.34	0.396596
$805$	16861.3	0.738238
$806$	26432.1	1.15512
$807$	−6991.11	−0.304955
$808$	25368.5	1.10453
$809$	19948.8	0.866950	0.433475	−	0.901166i	$-0.357287\pi$
$809$	19948.8	0.866950	0.433475	+	0.901166i	$0.357287\pi$
$810$	−3211.06	−0.139290
$811$	34609.1	1.49851	0.749253	−	0.662284i	$-0.230413\pi$
$811$	34609.1	1.49851	0.749253	+	0.662284i	$0.230413\pi$
$812$	−27736.8	−1.19873
$813$	12114.9	0.522616
$814$	−3635.80	−0.156554
$815$	−613.460	−0.0263663
$816$	824.064	0.0353529
$817$	−2060.06	−0.0882158
$818$	−13832.8	−0.591261
$819$	−60257.3	−2.57089
$820$	2998.50	0.127698
$821$	−1646.40	−0.0699874	−0.0349937	−	0.999388i	$-0.511141\pi$
$821$	−1646.40	−0.0699874	−0.0349937	+	0.999388i	$0.511141\pi$
$822$	2778.98	0.117917
$823$	−9215.19	−0.390305	−0.195153	−	0.980773i	$-0.562520\pi$
$823$	−9215.19	−0.390305	−0.195153	+	0.980773i	$0.562520\pi$
$824$	−7895.67	−0.333809
$825$	−546.942	−0.0230813
$826$	−10280.5	−0.433058
$827$	36509.8	1.53515	0.767576	−	0.640958i	$-0.221463\pi$
$827$	36509.8	1.53515	0.767576	+	0.640958i	$0.221463\pi$
$828$	15597.1	0.654635
$829$	−211.118	−0.00884492	−0.00442246	−	0.999990i	$-0.501408\pi$
$829$	−211.118	−0.00884492	−0.00442246	+	0.999990i	$0.501408\pi$
$830$	4637.53	0.193941
$831$	−14165.5	−0.591330
$832$	−15624.1	−0.651042
$833$	13544.1	0.563357
$834$	−8141.94	−0.338048
$835$	5493.23	0.227666
$836$	−1193.06	−0.0493576
$837$	18903.4	0.780642
$838$	−7930.86	−0.326930
$839$	−13822.3	−0.568770	−0.284385	−	0.958710i	$-0.591789\pi$
$839$	−13822.3	−0.568770	−0.284385	+	0.958710i	$0.591789\pi$
$840$	−5869.34	−0.241085
$841$	4796.27	0.196657
$842$	−4842.88	−0.198215
$843$	−6279.35	−0.256551
$844$	−26418.8	−1.07746
$845$	−31276.6	−1.27331
$846$	−2139.30	−0.0869394
$847$	−3441.46	−0.139610
$848$	−5685.70	−0.230245
$849$	−2725.17	−0.110162
$850$	1100.10	0.0443918
$851$	25888.3	1.04282
$852$	−6082.86	−0.244595
$853$	−29214.8	−1.17268	−0.586340	−	0.810065i	$-0.699432\pi$
$853$	−29214.8	−1.17268	−0.586340	+	0.810065i	$0.699432\pi$
$854$	−6376.02	−0.255483
$855$	−2189.21	−0.0875667
$856$	27008.5	1.07843
$857$	14061.7	0.560490	0.280245	−	0.959928i	$-0.409584\pi$
$857$	14061.7	0.560490	0.280245	+	0.959928i	$0.409584\pi$
$858$	3044.78	0.121150
$859$	−20991.3	−0.833775	−0.416888	−	0.908958i	$-0.636879\pi$
$859$	−20991.3	−0.833775	−0.416888	+	0.908958i	$0.636879\pi$
$860$	3094.66	0.122706
$861$	−5942.68	−0.235222
$862$	9529.64	0.376544
$863$	41800.7	1.64880	0.824398	−	0.566010i	$-0.191514\pi$
$863$	41800.7	1.64880	0.824398	+	0.566010i	$0.191514\pi$
$864$	−18671.3	−0.735198
$865$	−18136.1	−0.712884
$866$	23226.2	0.911382
$867$	−8090.79	−0.316929
$868$	−30835.5	−1.20579
$869$	8449.24	0.329828
$870$	2571.74	0.100218
$871$	73214.2	2.84819
$872$	−20388.8	−0.791803
$873$	−33336.3	−1.29240
$874$	−3410.23	−0.131982
$875$	3555.23	0.137358
$876$	8831.84	0.340640
$877$	4106.71	0.158123	0.0790615	−	0.996870i	$-0.474808\pi$
$877$	4106.71	0.158123	0.0790615	+	0.996870i	$0.474808\pi$
$878$	267.637	0.0102874
$879$	8769.61	0.336509
$880$	783.953	0.0300307
$881$	20412.4	0.780603	0.390301	−	0.920687i	$-0.372371\pi$
$881$	20412.4	0.780603	0.390301	+	0.920687i	$0.372371\pi$
$882$	−16253.9	−0.620518
$883$	46830.5	1.78479	0.892397	−	0.451251i	$-0.149022\pi$
$883$	46830.5	1.78479	0.892397	+	0.451251i	$0.149022\pi$
$884$	15255.6	0.580432
$885$	−2374.50	−0.0901896
$886$	−1795.40	−0.0680786
$887$	31519.8	1.19316	0.596579	−	0.802554i	$-0.296526\pi$
$887$	31519.8	1.19316	0.596579	+	0.802554i	$0.296526\pi$
$888$	−9011.63	−0.340552
$889$	−43154.9	−1.62809
$890$	6524.79	0.245743
$891$	−4666.64	−0.175464
$892$	27888.3	1.04683
$893$	−1165.18	−0.0436634
$894$	1759.77	0.0658338
$895$	−14950.9	−0.558385
$896$	35366.4	1.31865
$897$	−21680.0	−0.806996
$898$	−20306.7	−0.754613
$899$	32445.8	1.20370
$900$	3288.68	0.121803
$901$	−11595.3	−0.428740
$902$	−1749.35	−0.0645752
$903$	−6133.26	−0.226027
$904$	37227.4	1.36965
$905$	−4258.88	−0.156431
$906$	−7077.75	−0.259539
$907$	11712.7	0.428791	0.214396	−	0.976747i	$-0.431222\pi$
$907$	11712.7	0.428791	0.214396	+	0.976747i	$0.431222\pi$
$908$	−11554.5	−0.422303
$909$	28171.3	1.02792
$910$	−19791.6	−0.720974
$911$	−12306.0	−0.447546	−0.223773	−	0.974641i	$-0.571837\pi$
$911$	−12306.0	−0.447546	−0.223773	+	0.974641i	$0.571837\pi$
$912$	−538.629	−0.0195568
$913$	6739.74	0.244308
$914$	−24402.4	−0.883105
$915$	−1472.67	−0.0532075
$916$	−2320.79	−0.0837130
$917$	11646.7	0.419421
$918$	−4379.80	−0.157467
$919$	3741.96	0.134315	0.0671577	−	0.997742i	$-0.478607\pi$
$919$	3741.96	0.134315	0.0671577	+	0.997742i	$0.478607\pi$
$920$	12302.3	0.440865
$921$	3582.72	0.128181
$922$	−4578.86	−0.163554
$923$	−49257.3	−1.75658
$924$	−3552.02	−0.126464
$925$	5458.59	0.194030
$926$	−13995.6	−0.496680
$927$	−8767.98	−0.310656
$928$	−32047.5	−1.13363
$929$	24901.6	0.879434	0.439717	−	0.898136i	$-0.355079\pi$
$929$	24901.6	0.879434	0.439717	+	0.898136i	$0.355079\pi$
$930$	2859.05	0.100808
$931$	−8852.79	−0.311642
$932$	31022.8	1.09033
$933$	−7188.61	−0.252245
$934$	5724.70	0.200555
$935$	1598.78	0.0559204
$936$	−43965.0	−1.53530
$937$	−46031.4	−1.60489	−0.802444	−	0.596727i	$-0.796467\pi$
$937$	−46031.4	−1.60489	−0.802444	+	0.596727i	$0.796467\pi$
$938$	34287.1	1.19351
$939$	4140.74	0.143906
$940$	1750.36	0.0607346
$941$	9699.15	0.336008	0.168004	−	0.985786i	$-0.446268\pi$
$941$	9699.15	0.336008	0.168004	+	0.985786i	$0.446268\pi$
$942$	1369.14	0.0473555
$943$	12456.0	0.430143
$944$	3403.46	0.117344
$945$	−14154.4	−0.487240
$946$	−1805.45	−0.0620509
$947$	−6092.29	−0.209053	−0.104526	−	0.994522i	$-0.533333\pi$
$947$	−6092.29	−0.209053	−0.104526	+	0.994522i	$0.533333\pi$
$948$	8720.68	0.298770
$949$	71517.8	2.44633
$950$	−719.051	−0.0245569
$951$	−17866.4	−0.609209
$952$	17156.8	0.584091
$953$	−28868.6	−0.981265	−0.490633	−	0.871367i	$-0.663234\pi$
$953$	−28868.6	−0.981265	−0.490633	+	0.871367i	$0.663234\pi$
$954$	13915.1	0.472242
$955$	16526.8	0.559993
$956$	9497.69	0.321315
$957$	3737.51	0.126245
$958$	−10003.1	−0.337355
$959$	−26252.3	−0.883975
$960$	−1689.99	−0.0568169
$961$	6279.63	0.210789
$962$	−30387.5	−1.01843
$963$	29992.4	1.00363
$964$	−36155.9	−1.20799
$965$	6183.44	0.206271
$966$	−10153.0	−0.338166
$967$	−35960.7	−1.19588	−0.597941	−	0.801540i	$-0.704014\pi$
$967$	−35960.7	−1.19588	−0.597941	+	0.801540i	$0.704014\pi$
$968$	−2510.96	−0.0833732
$969$	−1098.47	−0.0364168
$970$	−10949.4	−0.362437
$971$	11212.2	0.370562	0.185281	−	0.982686i	$-0.440680\pi$
$971$	11212.2	0.370562	0.185281	+	0.982686i	$0.440680\pi$
$972$	−20157.2	−0.665168
$973$	76914.9	2.53420
$974$	−21616.2	−0.711118
$975$	−4571.27	−0.150151
$976$	2110.83	0.0692275
$977$	43762.3	1.43304	0.716520	−	0.697566i	$-0.245734\pi$
$977$	43762.3	1.43304	0.716520	+	0.697566i	$0.245734\pi$
$978$	369.395	0.0120777
$979$	9482.51	0.309563
$980$	13298.8	0.433485
$981$	−22641.4	−0.736884
$982$	−4526.43	−0.147092
$983$	2061.58	0.0668915	0.0334457	−	0.999441i	$-0.489352\pi$
$983$	2061.58	0.0668915	0.0334457	+	0.999441i	$0.489352\pi$
$984$	−4335.90	−0.140471
$985$	−3412.16	−0.110376
$986$	−7517.49	−0.242805
$987$	−3469.02	−0.111874
$988$	−9971.46	−0.321087
$989$	12855.5	0.413328
$990$	−1918.64	−0.0615943
$991$	−3986.63	−0.127790	−0.0638948	−	0.997957i	$-0.520352\pi$
$991$	−3986.63	−0.127790	−0.0638948	+	0.997957i	$0.520352\pi$
$992$	−35627.8	−1.14030
$993$	2610.33	0.0834203
$994$	−23067.8	−0.736082
$995$	−6283.36	−0.200197
$996$	6956.26	0.221303
$997$	−21988.4	−0.698474	−0.349237	−	0.937034i	$-0.613559\pi$
$997$	−21988.4	−0.698474	−0.349237	+	0.937034i	$0.613559\pi$
$998$	27631.4	0.876409
$999$	−21732.2	−0.688266

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.4.a.g.1.14	✓	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.4.a.g.1.14	✓	23	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1045 = 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1045.a (trivial)

\(f(q)\)	\(=\)	\(q+1.51379 q^{2} +1.98888 q^{3} -5.70843 q^{4} -5.00000 q^{5} +3.01075 q^{6} -28.4418 q^{7} -20.7517 q^{8} -23.0444 q^{9} +O(q^{10})\)	\(q+1.51379 q^{2} +1.98888 q^{3} -5.70843 q^{4} -5.00000 q^{5} +3.01075 q^{6} -28.4418 q^{7} -20.7517 q^{8} -23.0444 q^{9} -7.56896 q^{10} -11.0000 q^{11} -11.3534 q^{12} -91.9365 q^{13} -43.0550 q^{14} -9.94440 q^{15} +14.2537 q^{16} +29.0687 q^{17} -34.8844 q^{18} -19.0000 q^{19} +28.5422 q^{20} -56.5673 q^{21} -16.6517 q^{22} +118.567 q^{23} -41.2727 q^{24} +25.0000 q^{25} -139.173 q^{26} -99.5322 q^{27} +162.358 q^{28} -170.837 q^{29} -15.0537 q^{30} -189.923 q^{31} +187.591 q^{32} -21.8777 q^{33} +44.0039 q^{34} +142.209 q^{35} +131.547 q^{36} +218.344 q^{37} -28.7620 q^{38} -182.851 q^{39} +103.759 q^{40} +105.055 q^{41} -85.6311 q^{42} +108.424 q^{43} +62.7928 q^{44} +115.222 q^{45} +179.486 q^{46} +61.3255 q^{47} +28.3489 q^{48} +465.936 q^{49} +37.8448 q^{50} +57.8140 q^{51} +524.814 q^{52} -398.893 q^{53} -150.671 q^{54} +55.0000 q^{55} +590.216 q^{56} -37.7887 q^{57} -258.612 q^{58} +238.777 q^{59} +56.7669 q^{60} +148.090 q^{61} -287.503 q^{62} +655.423 q^{63} +169.944 q^{64} +459.683 q^{65} -33.1182 q^{66} -796.356 q^{67} -165.936 q^{68} +235.815 q^{69} +215.275 q^{70} +535.775 q^{71} +478.210 q^{72} -777.904 q^{73} +330.527 q^{74} +49.7220 q^{75} +108.460 q^{76} +312.860 q^{77} -276.798 q^{78} -768.112 q^{79} -71.2685 q^{80} +424.240 q^{81} +159.031 q^{82} -612.703 q^{83} +322.911 q^{84} -145.343 q^{85} +164.132 q^{86} -339.774 q^{87} +228.269 q^{88} -862.046 q^{89} +174.422 q^{90} +2614.84 q^{91} -676.831 q^{92} -377.733 q^{93} +92.8341 q^{94} +95.0000 q^{95} +373.096 q^{96} +1446.62 q^{97} +705.331 q^{98} +253.488 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q + 6 q^{2} + 9 q^{3} + 92 q^{4} - 115 q^{5} - 25 q^{6} - 37 q^{7} + 42 q^{8} + 170 q^{9}+O(q^{10}) \) 23 * q + 6 * q^2 + 9 * q^3 + 92 * q^4 - 115 * q^5 - 25 * q^6 - 37 * q^7 + 42 * q^8 + 170 * q^9	\( 23 q + 6 q^{2} + 9 q^{3} + 92 q^{4} - 115 q^{5} - 25 q^{6} - 37 q^{7} + 42 q^{8} + 170 q^{9} - 30 q^{10} - 253 q^{11} + 44 q^{12} - 37 q^{13} + 61 q^{14} - 45 q^{15} + 588 q^{16} - 73 q^{17} + 391 q^{18} - 437 q^{19} - 460 q^{20} - 127 q^{21} - 66 q^{22} - 175 q^{23} + 16 q^{24} + 575 q^{25} + 719 q^{26} + 21 q^{27} + 253 q^{28} + 71 q^{29} + 125 q^{30} + 302 q^{31} + 1107 q^{32} - 99 q^{33} + 1267 q^{34} + 185 q^{35} + 703 q^{36} - 500 q^{37} - 114 q^{38} + 457 q^{39} - 210 q^{40} + 770 q^{41} + 2596 q^{42} - 902 q^{43} - 1012 q^{44} - 850 q^{45} - 1101 q^{46} + 356 q^{47} + 1221 q^{48} + 908 q^{49} + 150 q^{50} - 451 q^{51} - 358 q^{52} + 1327 q^{53} + 2534 q^{54} + 1265 q^{55} + 3135 q^{56} - 171 q^{57} + 1014 q^{58} + 3619 q^{59} - 220 q^{60} - 1432 q^{61} + 1826 q^{62} + 1658 q^{63} + 4006 q^{64} + 185 q^{65} + 275 q^{66} - 605 q^{67} + 5128 q^{68} + 3099 q^{69} - 305 q^{70} + 3230 q^{71} + 2152 q^{72} - 637 q^{73} + 5063 q^{74} + 225 q^{75} - 1748 q^{76} + 407 q^{77} + 7230 q^{78} + 2074 q^{79} - 2940 q^{80} + 2291 q^{81} + 530 q^{82} + 3882 q^{83} + 5096 q^{84} + 365 q^{85} + 2262 q^{86} - 27 q^{87} - 462 q^{88} - 210 q^{89} - 1955 q^{90} + 4133 q^{91} - 6064 q^{92} + 824 q^{93} - 392 q^{94} + 2185 q^{95} + 2462 q^{96} + 2032 q^{97} + 7896 q^{98} - 1870 q^{99}+O(q^{100}) \) 23 * q + 6 * q^2 + 9 * q^3 + 92 * q^4 - 115 * q^5 - 25 * q^6 - 37 * q^7 + 42 * q^8 + 170 * q^9 - 30 * q^10 - 253 * q^11 + 44 * q^12 - 37 * q^13 + 61 * q^14 - 45 * q^15 + 588 * q^16 - 73 * q^17 + 391 * q^18 - 437 * q^19 - 460 * q^20 - 127 * q^21 - 66 * q^22 - 175 * q^23 + 16 * q^24 + 575 * q^25 + 719 * q^26 + 21 * q^27 + 253 * q^28 + 71 * q^29 + 125 * q^30 + 302 * q^31 + 1107 * q^32 - 99 * q^33 + 1267 * q^34 + 185 * q^35 + 703 * q^36 - 500 * q^37 - 114 * q^38 + 457 * q^39 - 210 * q^40 + 770 * q^41 + 2596 * q^42 - 902 * q^43 - 1012 * q^44 - 850 * q^45 - 1101 * q^46 + 356 * q^47 + 1221 * q^48 + 908 * q^49 + 150 * q^50 - 451 * q^51 - 358 * q^52 + 1327 * q^53 + 2534 * q^54 + 1265 * q^55 + 3135 * q^56 - 171 * q^57 + 1014 * q^58 + 3619 * q^59 - 220 * q^60 - 1432 * q^61 + 1826 * q^62 + 1658 * q^63 + 4006 * q^64 + 185 * q^65 + 275 * q^66 - 605 * q^67 + 5128 * q^68 + 3099 * q^69 - 305 * q^70 + 3230 * q^71 + 2152 * q^72 - 637 * q^73 + 5063 * q^74 + 225 * q^75 - 1748 * q^76 + 407 * q^77 + 7230 * q^78 + 2074 * q^79 - 2940 * q^80 + 2291 * q^81 + 530 * q^82 + 3882 * q^83 + 5096 * q^84 + 365 * q^85 + 2262 * q^86 - 27 * q^87 - 462 * q^88 - 210 * q^89 - 1955 * q^90 + 4133 * q^91 - 6064 * q^92 + 824 * q^93 - 392 * q^94 + 2185 * q^95 + 2462 * q^96 + 2032 * q^97 + 7896 * q^98 - 1870 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more